# Measuring epistasis in fitness landscapes: the correlation of fitness effects of mutations

## TL;DR

This paper introduces a new way to measure epistasis in fitness landscapes by analyzing the correlation of mutation effects.

## Contribution

A novel analytical measure of epistasis based on the correlation of fitness effects of mutations is proposed.

## Key findings

- The proposed measure captures mutation interactions and has a natural interpretation.
- The measure can be extended to landscapes with missing data or rank-only fitness information.
- The correlation of fitness effects depends on mutational distance, revealing patterns of epistasis.

## Abstract

Genotypic fitness landscapes are constructed by assessing the fitness of all possible combinations of a given number of mutations. In the last years, several experimental fitness landscapes have been completely resolved. As fitness landscapes are high-dimensional, simple measures of their structure are used as statistics in empirical applications. Epistasis is one of the most relevant features of fitness landscapes. Here we propose a new natural measure of the amount of epistasis based on the correlation of fitness effects of mutations. This measure has a natural interpretation, captures well the interaction between mutations and can be obtained analytically for most landscape models. We discuss how this measure is related to previous measures of epistasis (number of peaks, roughness/slope, fraction of sign epistasis, Fourier-Walsh spectrum) and how it can be easily extended to landscapes with missing data or with fitness ranks only. Furthermore, the dependence of the correlation of fitness effects on mutational distance contains interesting information about the patterns of epistasis. This dependence can be used to uncover the amount and nature of epistatic interactions in a landscape or to discriminate between different landscape models.